Design and Stabilization of Sampled-Data Neural-Network-Based ...
Proceedings of International Joint Conference on Neural Networks, Montreal, Canada, July 31 - August 4, 2005
Stabilization of Sampled-Data NeuralNetwork-B ased Control Systems'
Design and
H.K. Lam and F.H.F. Leung Centre for Multimedia Signal Processing, Department of Electronic and Information Engineering, The Hong Kong Polytechnic
University, Hung Hom, Kowloon, Hong Kong
Abstract - This paper presents the design and stability analysis of sampled-data neural-network-based control systems. A continuous-time nonlinear plant and a sampled-data three-layer fully-connected feed-forward neural-network-based controller are connected in a closed-loop to perform a control task. Stability conditions will be derived to guarantee the closed-loop system stability. Linear-matrix-inequality- and geneticalgorithm-based approaches will be employed to obtain the maximum sampling period and connection weights of the neural network subject to the considerations of the system stability and performance. An application example will be given to illustrate the design procedure and effectiveness of the proposed approach.
1. INTRODUCTION The superior learning and generalization abilities of neural networks have attracted the public attention for many years. It was shown that a three-layer fully-connected feed-forward neural-network (TLFCFFNN) is a universal approximator which is able to approximate any smooth continuous function in a compact domain to an arbitrary accuracy [1]. Because of this outstanding property, neural networks were widely applied in different applications to handle different problems such as forecasting, handwritten character recognitions, control problems, etc. This paper focuses on the system stability and performance issues of the neural-network-based sampled-data control problems. A neural-network-based control system is composed of a nonlinear plant and a neural-network-based controller connected in closed-loop. The nonlinearities of the plant and neural network, and the complexity of the network structure make the analysis work difficult and complex. Different neural-network-based control approaches subject to the consideration of the system stability were proposed. In [2], an adaptive neural-network based controller with variable hidden nodes was presented. The stability of the closed-loop system is achieved through on-line parameter adaptation, which is governed by adaptive laws derived based on the Lyapunov stability theory. The main idea of this approach is to compensate the nonlinearity of the plant by making use of the on-line estimated parameter values. The estimation error is a potential component to cause the instability of the closedloop system. To handle the effect of the estimation error to the system stability, adaptive neural-network based The work described in this paper was fully supported by a grant from The Hong Kong Polytechnic University (Project No. G-YX3 1).
0-7803-9048-2/05/$20.00 02005 IEEE
controllers [3]-[6] with switching control signals were proposed. These switching signals may introduce a chattering effect to the system. In [7]-[8], adaptive neural networks combined with other conventional controllers were proposed. In most of these approaches, the use of the neural networks is mainly for modeling the unknown nonlinearity of the plants. Another control schemes were then employed to achieve the system stability by compensating the system nonlinearity based on the learnt information of the neural networks. In summary, the system stability was achieved by adaptive and/or sliding mode control techniques in most of these approaches but not by the neural network itself. These approaches require that the network parameters are on-line changing according to some adaptive laws. This requirement will increase the computational demand, structural complexity, and implementation cost of the neural-networkbased controller. In [9]-[1 1], stability conditions have been derived for a class of neural-network-based control systems with a feed-forward multilayer-perceptron (MLP) neural network. The derived stability conditions were only for checking the stability of the neural-network-based control systems. However, the ways for finding the network parameters and optimizing the system performance were ignored. These are in fact two important issues for putting the neural-network-based controller into practice. In most of the published work, the efforts were put to purely continuous-time or discrete-time neural-networkbased control systems. The sampled-data neural-networkbased control systems are seldom considered. Fig. 1 shows the sampled-data TLFCFFNN-based control system, which is formed by a continuous-time nonlinear plant and a sampleddata TLFCFFNN-based controller connected in closed-loop. Referring to this figure, the sampled-data TLFCFFNN-based controller is formed by a sampler, a discrete-time TLFCFFNN-based controller and a zero-order-hold (ZOH) unit. The sampler takes the system states every h seconds to the discrete-time TLFCFFNN-based controller for generating the control signals. The constant time period h is called the sampling period. The control signals generated by the discrete-time TLFCFFNN-based controller will be fed to the ZOH unit to give the control signals to the nonlinear plant to realize the control process. The control signals from the ZOH unit are held constant during the sampling period. It can be seen that the introduction of the sampler and ZOH unit make the dynamics of the closed-loop system more complex, which increases the difficulty of the system analysis.
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However, under this control framework, the development and implementation costs may be reduced as the sampled-data TLFCFFNN-based controller can be realized by a microcomputer or a digital computer. In this paper, stability conditions will be derived to guarantee the stability of the TLFCFFNN-based control systems using the Lyapunov approach. The derived stability conditions will be employed to aid the design of stable sampled-data TLFCFFNN-based controllers. The finding of the maximum sampling period and connection weights of the TLFCFFNN-based controllers, and the optimization of the system performance subject to the system stability are formulated as a generalized eigenvalue minimization problem (GEVP) [12] and genetic algorithm (GA) [ 13] minimization problem respectively. These problems can be solved numerically by convex programming techniques [ 12] and GA [ 13].
x(ty) = [x,(t7) x2(tY)
x,(t5)Jj denotes the sampled system state vector x(t) at the sampled time ty. The sampleddata TLFCFFNN-based controller for the nonlinear system of (2), with nflou = mn, is defined as, yY (ty) XI(ty) Y, (t5) Y2 (ty) * Y2nn (ty) X2 (ty) Yn+2 (ty) Yn+1(ty)
2. NONLINEAR SYSTEM AND NEURAL-NETwORK-BASED CONTROLLER A TLFCFFNN-based control system as shown in Fig. 1 consists of a continuous-time nonlinear system and a
TLFCFFNN-based controller.
...
... Ymn (ty) Y(m-l)n+2 (ty)+ , tf , m,j xi (ty ) + b,
u()= Y(m-I)n+l (tY) ,
nh
n
Xn (ty) .t I .)f -
(5)
t < t~+l
It should be noted that u(t) = u(tr) holds a constant vector value by a ZOH unit for ty < t < t,1. From (4) and (5), we have, *-- g g(2I, g ..+2. x,(ty) .
n,,
n
IZtfl Emj ixi (tr) + bj j=I
I
u(t)
'k='
9(m,-I),n+I.j 9(.-I)ti+2,j
",
=
Ith
..
gimi
_Xn (ty )
it
I t f YEi=l mij xi (ty) + b,
1=1
A. Nonlinear System The continuous-time nonlinear system has the following
Emj (x(t))Gjx(ty)
=
where
(2) x(t) = i=1, wi (x(t))(A1x(t) + B1u(t)) where A1 e 9VIx" and B. E 9i xrm are the constant system and
mj(X(t7))
input matrices respectively; p is a nonzero positive integer;
wi(x(t)) has the following properties: Xw,(x(t)) = , wi (x(t)) E [O i], i = 1, 2, ..., p (3) i=l It should be noted that the value of wi(x(t)) is unknown if
the nonlinear system is subject to parameter uncertainties.
B. Sampled-Data Three-Layer Fully-Connected FeedForward Neural-Network-Based Controller The input-output relationship of a discrete-time TLFCFFNN [ 14] is defined as, nh
n
Yk(t7) = Egkj tf' mJJxi(ty)+bj , k= 1, 2, ..., nol (4) where t,= )h, y= 0, 1, 2, ..., denotes a sampled instance; h =
t,i - tydenotes the constant sampling period; mji denotes the connection weight between the j-th hidden node and the i-th input node; gkj denotes the connection weight between the kth output node and the j-th hidden node; bj denotes the bias for the j-th hidden node; t1(.) denotes the activation function; noul denotes the number of output nodes;
(6)
j=l
form: (1) x(t) = f (x(t), u(t)) where x(t) = [xl(t) x2(t) * x,,(t)Y is the system state - Um (t)]T is the input vector; vector; u(t) = [u, (t) U2 (t) f(-) denotes a nonlinear function with a known form. It is assumed that the nonlinear system of (1) can be written as,
92,j
...
gl,
gfn+2,j
*--
92n,j
gl.i gGr=-),+I,j 9(m-I)ii+l,j =
9(m-l)ii+2,j
tf ( mj ix, (tv) + b)
,,
(7)
g. .g
KE
[o
ZEtf E m.ixi(ti) + b)
1]
(8)
l,h
which has the property that Z mj(x(ty)) = 1 . It is assumed J=1
that the activation function t1() is chosen such that
tf (mj ixl(t)+bj) >O and
Ztf (m,.jxj(tY)+bj JO
at
any time to satisfy the property of (8).
C. Sampled-Data TLFCFFNN-Based Control Systems A sampled-data TLFCFFNN-based control system as shown in Fig. 1 is formed by connecting the continuous-time nonlinear system of (2) and the sampled-data TLFCFFNNbased controller of (6) in closed-loop. In the following, wi(x(t)) and mj(x(tr)) are written as wi and m1 respectively. From (2) and (6), and with the property that wI = EMi = j=1
nh
E=E wi i=l j=l
=
1,
we have,
+ x(t)= i=1 J=Ewi, tmj(Aix(t) BiG1x(t7))
Let
z(t) = t - ty,< h for t,< t < t,,,. From (9), we have,
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(9)
X(t) = i i w,mi (Aix(t) + BiGjx(t - r(t)))
(10)
i=l j=l
III. STABILrrY AND DESIGN OF SAMPLE-DATA TLFCFFNNBASED CONTROL SYSTEMS The stability, design and performance optimization of the sampled-data TLFCFFNN-based control system of (10) will be investigated in this section.
A. Stability Analysis and Maximum Sampling Period Consider the following Lyapunov function candidate to investigate the stability of the sampled-data TLFCFFNNbased control system, V(t) = V1(t) + V2(t) + VA3(t) (11) where V1(t) = x(t)TPx(t) (12) (13) '2 (t) = e,h f+a (V)TRi(V)d4da
V3 (t) =r()X(9)j Sx(V)d9
(14)
=hJ>w,mj[A, BiGI x(t) - fh(qp) i=I j=Ik1IRi(,p)d(p
0 as t -4 oo > x(t) -0 as t -+ oo. By Schur complement [12], Qij < 0 is X>O, y>0,Mi >0, M>0, Smj 2X-M] equivalent to the following LMI, +Y R + S + hR.. PBiG .-R.i mii ° hAiT RTI XA1 A[T +A,X +Mij+MJT AiTP+ PA,GT+RTi+iRT ' T T N TBT o 0 N TBT hI G B PIR.j -S hG B R < O
[hRAi
hRBiG1
LAiX
-hR
Pre-multiply and post-multiply diag{P-1, P-, R-1 to the above equation and let X = P , M,1 = P RjjP-1 Y = P-'SP-', M..= P-'R jP-', M=R-' and Gj=NjX-', we have,
XAiT + AiX+Mij+Mi +Y+hMi BiNj-Mij NTB T_ M
NiX
T
BhBiNj-Y
hBiNJ
hAiX
From (15), it is required that ,i ii and post-multiply
> [PRJPI
hXA T
-hM
'j .20. Pre-multiply
R1. 0, we have, RT diag(P-1, P-')to FR. i] . P'RW'
1. 0
>j.0
=> [AYT
(26)
It should be noted that (26) is not an LMI. Based on the following property, (26) can be represented as an LMI. Consider the following inequality, (X-M)TM-'(X-M)= XTM'X-XT -X+M > 0 > XM-'X > 2X-M (27) From (26) and (27), it can be seen that
FMA
[M,i
NI,
[r0 2XT
iM
1
XM-X1
2
M]
M,1 1
MiT 2X - M .0
M,T XM'j Mjlij
implies
-
Hence,
. >.0.
The stability conditions in terms of linear matrix inequalities (LMIs) [12], the connections weights of the TLFCFFNN and the maximum sampling period can be determined by solving the generalized eigenvalue minimization problem (GEVP) as stated in the following theorem.
Theorem 1: The sampled-data TLFCFFNN-based control system of (10) formed by the continuous-time nonlinear system in form of (2) and the sampled-data TLFCFFNN-
BiNj
-M
B,N;-Mij -Y
O
L
0
O 01
where the connection weights of the TLFCFFNN are defined Gi = NJX- . The maximum sampling period h is
as
obtained by minimizing - subject to the above LMIs. X,Y,M,j,M,j,M,Nj
h
B. Tuning of mji and bj In Theorem 1, the maximum sampling period and the connection weight gkj are determined based on the LMI approach. In this section, the connection weight mji and the bias bj will be determined. Owning to the nonlinear nature of the activation function of the TLFCFFNN, it is difficult to formulate the finding of mji and bj into an LMI problem. Instead, GA can be employed to tune the values of mji and bj by minimizing the following performance index,
.o[u(t) [J 2T J 3 Ju(t)J"
(28)
It should be noted that the values of the sampling period h and the connection weight gkj are kept constant during the turning process. As the system stability is determined by the sampling period and the connection weight gkj only, the values of mji and bj can be freely tuned to optimize the system performance. During the tuning process, mji is the gene to form the chromosome of the GA process.
C. Design Procedure The design procedure of the sampled-data TLFCFFNNbased controller is given as follows. Step I) Obtain the model of the nonlinear plant in the form of (2).
Step II) Determine the number of hidden node nh and the activation functions for the sampled-data TLFCFFNN-based controller in the form of (6). Step III) Determine the maximum sampling period, hm., and the connection weight gkj by Theorem 1. Choose a constant sampling period 0 < h < hma.x Step IV) Under the chosen sampling period h and the connection weight gkj, obtain the connection weight mji and the bias bj by the GA process. Realize the sampled-data TLFCFFNN-based controller based on the determined h, gkj, mji and bj. IV APPLICATION EXAMPLE
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The proposed sampled-data TLFCFFNN-based controller will be employed to stabilize an inverted pendulum subject to parameter uncertainties. The objective is to drive all the system states of the inverted pendulum to zero at the steady state.
pendulum. From (6), the sampled-data TLFCFFNN-based controller is given by, 4 u(t) = E2mj(x(t,))G jx(t7) , t, < t < t)
The system behaviour of the inverted Step I) pendulum is described by the following dynamic equation. g sin(9(t)) - ampL.9(t)2 sin(20(t)) /2 - a cos(9(t))u(t) 4L/3 - ampLcos2(6(t))
transfer function, i.e, tf (
(29)
6(t) is the angular displacement of the pendulum, g = 9.8m/s2 is the acceleration due to gravity, mp E [mp mp ] = [2 5]kg is the mass of the pendulum, M, E [Memin Mc, ] = [30 35]kg is the mass of the cart, a = 11(mp + MJ), 2L = 1 m is the length of the pendulum, and u(t) is the where
force applied to the cart. mp and MC contain parameter uncertainties of the system. The inverted pendulum subject to parameter uncertainties can be represented by the following model, 4 x(t) = E w; (Aix(t) + Biu(t)) (30) ,=,
XI (t)E
[x
xl ]
= [-
f2(x(t))
Al =A2=
0
B, =B3=Lf
fl
=
-
1
4L13-am Lcos( 2(X) 4L/3-amALcos2 (X1
B2= B4
16.4640, f2.
(t)E
ampLcos2(xl (t)
=
f[fj
XI
;
(t))
f2.
;
= -0.0492;
=E WMI~(AZx(t)))XflM, (f2(X(t)))) (xW wi~~~~~~It 4
flM(l,(f(x(t)))=- -f(x(0t)+f, M
(A, (X(t)))
=
~(rx(t))) =
flMw(W2(x(t))) =
1-fM (t ) (X(t))) -f2
1-
((t
)) +
MI(
,W for for
for
2(x(t))) for
K
=
=
1,
2;
=
3,
4;
3
and
1,
+
Step III) The maximum sampling period, hm.,, is found to be 0.0284s based on Theorem 1. A constant sampling period 0 < h = 0.02 < h,. is chosen for the sampled-data TLFCFFNN-based controller. MATALB LMI tool box is employed to solve the solution of the GEVP in Theorem 1 numerically. The connection weights, G1 = N1X-', are obtained as GI = [g1j1 g2,1] = [1358.9268 347.8515], G2 = [g1,2 g22] = G3 = [g113 g2,3] = [1319.7667 337.4474] and G4 = [g114 g924] = [1291.0126 329.7995]. It can be concluded from Theorem 1 that the optimized sampled-data TLFCFFNN-based control system is asymptotically stable. Step IV) The parameters mji and bj of the activation function
~~~~~~~~~1 can be obtained by
2
(ty) +bj?J= 2mj.,rj , (ty)+bj) tfQlmjjxi(i=l~~~~~~~~~~Y )
)) ( |~~~~(t) )
=11.3533 and
-0.0192 and
=
optimizing the following performance index using improved GA [16] under m, = 2kg and Mp = 30kg.
LX2,X2,,,n
0 1 fi0
A3=A4
and
and
x
3
3
4L/3-
and
and
-
mjjxj (t7) + bj
l+e4i='
x(t) = [x1 (t) x2 ()] = [9(t) @(t)]T
where
j=I
where the logarithmic sigmoid function is employed as the
9=
2, 4.
Step II) A sampled-data TLFCFFNN-based ci ontroller with four hidden nodes is employed to handle the inverted
2J JX[(t) dt (31) The values of the connection weight and bias, mji and bj, before and after GA process are listed in Table I. The obtained sampled-data TLFCFFNN-based controller is employed to handle the inverted pendulum subject to parameter uncertainties. Under the initial state conditions of f
[ut)
x(0) =
3
I
T
0l IT
x(0) = x)
x(0)
6J~ of6()[-
0 ]
and n
x(0) = [-
] with m, = 2kg and Mp = 30kg, the system
responses
and control signals of the sampled-data
TLFCFFNN-based control systems are shown in Fig. 2. Referring to these figures, it can be seen that the proposed sampled-data TLFCFFNN-based controller is able to stabilize the- inverted pendulum. The proposed sampled-data TLFCFFNN-based controller can be easily implemented by a microcontroller or a digital computer. V. CONCLUSION A sampled-data TLFCFFNN-based controller, which is formed by a sampler, a TLFCFFNN and a ZOH unit, has been proposed for continuous-time nonlinear systems. Based on the Lyapunov-based approach, the stability of the sampled-data TLFCFFNN-based control systems has been investigated. Stability conditions have been derived to guarantee the stability of the sampled-data TLFCFFNN-
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based control systems respectively. The findings of the maximum sampling period and the network connection weights, and the optimization of system performance have been formulated as generalized eigenvalue and genetic algorithm minimization problems. An application example on stabilizing an inverted pendulum subject to parameter uncertainties has been given to illustrate the design procedure and the effectiveness of the proposed approach.
1.5
O
REFERENCES
[1] B. Widrow and M.A. Lehr, "30 years of adaptive neural networks: Perceptron, madaline, and backpropagation," Proceedings of the IEEE, vol. 78, no. 9, Sept. 1990, pp. 1415-1442. [2] G.P. Liu, V. Kadirkamanathan, and S.A. Billings, "Variable neural networks for adaptive control of nonlinear systems," IEEE Transactions on System, Man, and Cybernetic - Part C: Applications and Reviews, vol. 29, no. 1, pp. 3443, Feb. 1999. [3] H.D. Patinto, R. Carelli, and B.R. Kuchen, "Neural network for advanced control of robot manipulators," IEEE Transactions on Neural Networks, vol. 13, no.2, pp. 343-354, March 2002. [4] R.M. Sanner and J.J.E. Slotine, "Gaussian networks for direct adaptive control," IEEE Transactions on Neural Networks, vol. 3, no. 6, pp. 837-863, Nov. 1992. [5] R. Carelli, E.F. Camacho and D. Patifio, "A neural network based feedforward adaptive controller for robots," IEEE Transactions on Systems, Man, and Cybernetics, vol. 25, no. 9, pp. 1281-1288, Sept. 1995. [6] C.L. Hwang, "Neural-network-based variable structure control of electrohydralic servosystems subject to huge uncertainties without persistent excitation," IEEE Transactions on Mechatronics, vol. 4, no. 1, pp. 50-59, March 1999. [7] S. Lin and A.A. Goldenberg, "Neural-network control of mobile manipulators," IEEE Transactions on Neural Networks, vol. 12, no. 5, pp. 1121-1133, Sept. 2001. [8] C.L. Lin, "Control of perturbed systems using neural networks," IEEE Transactions on Neural Networks, vol. 9, no. 5, pp. 1046-1050, Sept. 1998. [9] K. Tanaka, "Stability and stabilizability of fuzzy-neural-linear control systems," IEEE Transactions on Fuzzy Systems, vol. 3, no. 4, pp. 438447, Nov. 1995. [10] K. Tanaka, "An approach to stability criteria of neural-network control systems," IEEE Transactions on Neural Network, vol. 7, no. 3, pp. 629-642, May 1996. [11] J.D. Hwang and F.H. Hsiao, "Stability analysis of neural network interconnected systems," IEEE Transactions on Neural Networks, vol. 14, no. 1, pp. 201-208, Jan. 2003. [12] S. Boyd, L. El. Ghaoui, E. Feron, and V. Balakrishnan, Linear matrix inequalities in system and control theory. SLAM, Philadelphia, 1994. [13] Z. Michalewicz, Genetic Algorithm + Data Structures = Evolution Programs, 2nd Ed. Springer-Verlag, 1994. [14] F.M. Ham and I. Kostanic, Principles of Neurocomputing for Science & Engineering, McGraw Hill, 2001. [15] B.D.O. Anderson and J.B. Moore, Optimal Control: Linear Quadratic Methods. Prentice Hall, 1990. [16] F.H.F. Leung, H.K. Lam, S.H. Uing and P.K.S. Tam, "Tuning of the structure and parameters of neural network using an improved genetic algorithm," IEEE Trans. Neural Networks, vol. 14, no. 1, pp. 79-88, Jan. 2003.
-1.5I 0 0.2 0.4 0.6 0.8
1 1.2 Time (sec)
1.6
1.8
2
1.4
1.6
1.8
2
Fig. 2(a). xl(t).
Fig. 2(b). x2(t).
0
0.2
0.4
0.6
0.8
1 1.2 Tim (s.c)
Fig. 2(c). u(t) Fig. 2. System responses and control signals of the inverted pendulum with the proposed sampled-data TLFCFFNN-based controller for mp = 2kg and
MC = 30kg.
TABLE 1. THE VALUES OF THE CONNECTION WEIGHT AND BIAS, M,,, AND Bi,
Parameter ml X Ml.2
bi
M2.1
M2.2
Continuous-time |
1.4
b2
tINonlinear Plant |I*xt
M31
M3.2 b3
M4_ 1
1742 _
_
BEFORE AND AFTFR THE (',A PRAFq.C1
Initial parameter values -0.0591
0.5511 --0.2765 --0.9863 0.5118 -0.3234 0.3058 0.1582 -0.7899 _-0.6254 -0.7569 0.4423
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Parameter values after GA process -0.7283 -0.7605 -0.2638 0.8534 -0.8836 0.9208
0.9816 0.9656 0.8918 0.9972 0.9993 0.8240
Stabilization of Sampled-Data NeuralNetwork-B ased Control Systems'
Design and
H.K. Lam and F.H.F. Leung Centre for Multimedia Signal Processing, Department of Electronic and Information Engineering, The Hong Kong Polytechnic
University, Hung Hom, Kowloon, Hong Kong
Abstract - This paper presents the design and stability analysis of sampled-data neural-network-based control systems. A continuous-time nonlinear plant and a sampled-data three-layer fully-connected feed-forward neural-network-based controller are connected in a closed-loop to perform a control task. Stability conditions will be derived to guarantee the closed-loop system stability. Linear-matrix-inequality- and geneticalgorithm-based approaches will be employed to obtain the maximum sampling period and connection weights of the neural network subject to the considerations of the system stability and performance. An application example will be given to illustrate the design procedure and effectiveness of the proposed approach.
1. INTRODUCTION The superior learning and generalization abilities of neural networks have attracted the public attention for many years. It was shown that a three-layer fully-connected feed-forward neural-network (TLFCFFNN) is a universal approximator which is able to approximate any smooth continuous function in a compact domain to an arbitrary accuracy [1]. Because of this outstanding property, neural networks were widely applied in different applications to handle different problems such as forecasting, handwritten character recognitions, control problems, etc. This paper focuses on the system stability and performance issues of the neural-network-based sampled-data control problems. A neural-network-based control system is composed of a nonlinear plant and a neural-network-based controller connected in closed-loop. The nonlinearities of the plant and neural network, and the complexity of the network structure make the analysis work difficult and complex. Different neural-network-based control approaches subject to the consideration of the system stability were proposed. In [2], an adaptive neural-network based controller with variable hidden nodes was presented. The stability of the closed-loop system is achieved through on-line parameter adaptation, which is governed by adaptive laws derived based on the Lyapunov stability theory. The main idea of this approach is to compensate the nonlinearity of the plant by making use of the on-line estimated parameter values. The estimation error is a potential component to cause the instability of the closedloop system. To handle the effect of the estimation error to the system stability, adaptive neural-network based The work described in this paper was fully supported by a grant from The Hong Kong Polytechnic University (Project No. G-YX3 1).
0-7803-9048-2/05/$20.00 02005 IEEE
controllers [3]-[6] with switching control signals were proposed. These switching signals may introduce a chattering effect to the system. In [7]-[8], adaptive neural networks combined with other conventional controllers were proposed. In most of these approaches, the use of the neural networks is mainly for modeling the unknown nonlinearity of the plants. Another control schemes were then employed to achieve the system stability by compensating the system nonlinearity based on the learnt information of the neural networks. In summary, the system stability was achieved by adaptive and/or sliding mode control techniques in most of these approaches but not by the neural network itself. These approaches require that the network parameters are on-line changing according to some adaptive laws. This requirement will increase the computational demand, structural complexity, and implementation cost of the neural-networkbased controller. In [9]-[1 1], stability conditions have been derived for a class of neural-network-based control systems with a feed-forward multilayer-perceptron (MLP) neural network. The derived stability conditions were only for checking the stability of the neural-network-based control systems. However, the ways for finding the network parameters and optimizing the system performance were ignored. These are in fact two important issues for putting the neural-network-based controller into practice. In most of the published work, the efforts were put to purely continuous-time or discrete-time neural-networkbased control systems. The sampled-data neural-networkbased control systems are seldom considered. Fig. 1 shows the sampled-data TLFCFFNN-based control system, which is formed by a continuous-time nonlinear plant and a sampleddata TLFCFFNN-based controller connected in closed-loop. Referring to this figure, the sampled-data TLFCFFNN-based controller is formed by a sampler, a discrete-time TLFCFFNN-based controller and a zero-order-hold (ZOH) unit. The sampler takes the system states every h seconds to the discrete-time TLFCFFNN-based controller for generating the control signals. The constant time period h is called the sampling period. The control signals generated by the discrete-time TLFCFFNN-based controller will be fed to the ZOH unit to give the control signals to the nonlinear plant to realize the control process. The control signals from the ZOH unit are held constant during the sampling period. It can be seen that the introduction of the sampler and ZOH unit make the dynamics of the closed-loop system more complex, which increases the difficulty of the system analysis.
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However, under this control framework, the development and implementation costs may be reduced as the sampled-data TLFCFFNN-based controller can be realized by a microcomputer or a digital computer. In this paper, stability conditions will be derived to guarantee the stability of the TLFCFFNN-based control systems using the Lyapunov approach. The derived stability conditions will be employed to aid the design of stable sampled-data TLFCFFNN-based controllers. The finding of the maximum sampling period and connection weights of the TLFCFFNN-based controllers, and the optimization of the system performance subject to the system stability are formulated as a generalized eigenvalue minimization problem (GEVP) [12] and genetic algorithm (GA) [ 13] minimization problem respectively. These problems can be solved numerically by convex programming techniques [ 12] and GA [ 13].
x(ty) = [x,(t7) x2(tY)
x,(t5)Jj denotes the sampled system state vector x(t) at the sampled time ty. The sampleddata TLFCFFNN-based controller for the nonlinear system of (2), with nflou = mn, is defined as, yY (ty) XI(ty) Y, (t5) Y2 (ty) * Y2nn (ty) X2 (ty) Yn+2 (ty) Yn+1(ty)
2. NONLINEAR SYSTEM AND NEURAL-NETwORK-BASED CONTROLLER A TLFCFFNN-based control system as shown in Fig. 1 consists of a continuous-time nonlinear system and a
TLFCFFNN-based controller.
...
... Ymn (ty) Y(m-l)n+2 (ty)+ , tf , m,j xi (ty ) + b,
u()= Y(m-I)n+l (tY) ,
nh
n
Xn (ty) .t I .)f -
(5)
t < t~+l
It should be noted that u(t) = u(tr) holds a constant vector value by a ZOH unit for ty < t < t,1. From (4) and (5), we have, *-- g g(2I, g ..+2. x,(ty) .
n,,
n
IZtfl Emj ixi (tr) + bj j=I
I
u(t)
'k='
9(m,-I),n+I.j 9(.-I)ti+2,j
",
=
Ith
..
gimi
_Xn (ty )
it
I t f YEi=l mij xi (ty) + b,
1=1
A. Nonlinear System The continuous-time nonlinear system has the following
Emj (x(t))Gjx(ty)
=
where
(2) x(t) = i=1, wi (x(t))(A1x(t) + B1u(t)) where A1 e 9VIx" and B. E 9i xrm are the constant system and
mj(X(t7))
input matrices respectively; p is a nonzero positive integer;
wi(x(t)) has the following properties: Xw,(x(t)) = , wi (x(t)) E [O i], i = 1, 2, ..., p (3) i=l It should be noted that the value of wi(x(t)) is unknown if
the nonlinear system is subject to parameter uncertainties.
B. Sampled-Data Three-Layer Fully-Connected FeedForward Neural-Network-Based Controller The input-output relationship of a discrete-time TLFCFFNN [ 14] is defined as, nh
n
Yk(t7) = Egkj tf' mJJxi(ty)+bj , k= 1, 2, ..., nol (4) where t,= )h, y= 0, 1, 2, ..., denotes a sampled instance; h =
t,i - tydenotes the constant sampling period; mji denotes the connection weight between the j-th hidden node and the i-th input node; gkj denotes the connection weight between the kth output node and the j-th hidden node; bj denotes the bias for the j-th hidden node; t1(.) denotes the activation function; noul denotes the number of output nodes;
(6)
j=l
form: (1) x(t) = f (x(t), u(t)) where x(t) = [xl(t) x2(t) * x,,(t)Y is the system state - Um (t)]T is the input vector; vector; u(t) = [u, (t) U2 (t) f(-) denotes a nonlinear function with a known form. It is assumed that the nonlinear system of (1) can be written as,
92,j
...
gl,
gfn+2,j
*--
92n,j
gl.i gGr=-),+I,j 9(m-I)ii+l,j =
9(m-l)ii+2,j
tf ( mj ix, (tv) + b)
,,
(7)
g. .g
KE
[o
ZEtf E m.ixi(ti) + b)
1]
(8)
l,h
which has the property that Z mj(x(ty)) = 1 . It is assumed J=1
that the activation function t1() is chosen such that
tf (mj ixl(t)+bj) >O and
Ztf (m,.jxj(tY)+bj JO
at
any time to satisfy the property of (8).
C. Sampled-Data TLFCFFNN-Based Control Systems A sampled-data TLFCFFNN-based control system as shown in Fig. 1 is formed by connecting the continuous-time nonlinear system of (2) and the sampled-data TLFCFFNNbased controller of (6) in closed-loop. In the following, wi(x(t)) and mj(x(tr)) are written as wi and m1 respectively. From (2) and (6), and with the property that wI = EMi = j=1
nh
E=E wi i=l j=l
=
1,
we have,
+ x(t)= i=1 J=Ewi, tmj(Aix(t) BiG1x(t7))
Let
z(t) = t - ty,< h for t,< t < t,,,. From (9), we have,
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(9)
X(t) = i i w,mi (Aix(t) + BiGjx(t - r(t)))
(10)
i=l j=l
III. STABILrrY AND DESIGN OF SAMPLE-DATA TLFCFFNNBASED CONTROL SYSTEMS The stability, design and performance optimization of the sampled-data TLFCFFNN-based control system of (10) will be investigated in this section.
A. Stability Analysis and Maximum Sampling Period Consider the following Lyapunov function candidate to investigate the stability of the sampled-data TLFCFFNNbased control system, V(t) = V1(t) + V2(t) + VA3(t) (11) where V1(t) = x(t)TPx(t) (12) (13) '2 (t) = e,h f+a (V)TRi(V)d4da
V3 (t) =r()X(9)j Sx(V)d9
(14)
=hJ>w,mj[A, BiGI x(t) - fh(qp) i=I j=Ik1IRi(,p)d(p
0 as t -4 oo > x(t) -0 as t -+ oo. By Schur complement [12], Qij < 0 is X>O, y>0,Mi >0, M>0, Smj 2X-M] equivalent to the following LMI, +Y R + S + hR.. PBiG .-R.i mii ° hAiT RTI XA1 A[T +A,X +Mij+MJT AiTP+ PA,GT+RTi+iRT ' T T N TBT o 0 N TBT hI G B PIR.j -S hG B R < O
[hRAi
hRBiG1
LAiX
-hR
Pre-multiply and post-multiply diag{P-1, P-, R-1 to the above equation and let X = P , M,1 = P RjjP-1 Y = P-'SP-', M..= P-'R jP-', M=R-' and Gj=NjX-', we have,
XAiT + AiX+Mij+Mi +Y+hMi BiNj-Mij NTB T_ M
NiX
T
BhBiNj-Y
hBiNJ
hAiX
From (15), it is required that ,i ii and post-multiply
> [PRJPI
hXA T
-hM
'j .20. Pre-multiply
R1. 0, we have, RT diag(P-1, P-')to FR. i] . P'RW'
1. 0
>j.0
=> [AYT
(26)
It should be noted that (26) is not an LMI. Based on the following property, (26) can be represented as an LMI. Consider the following inequality, (X-M)TM-'(X-M)= XTM'X-XT -X+M > 0 > XM-'X > 2X-M (27) From (26) and (27), it can be seen that
FMA
[M,i
NI,
[r0 2XT
iM
1
XM-X1
2
M]
M,1 1
MiT 2X - M .0
M,T XM'j Mjlij
implies
-
Hence,
. >.0.
The stability conditions in terms of linear matrix inequalities (LMIs) [12], the connections weights of the TLFCFFNN and the maximum sampling period can be determined by solving the generalized eigenvalue minimization problem (GEVP) as stated in the following theorem.
Theorem 1: The sampled-data TLFCFFNN-based control system of (10) formed by the continuous-time nonlinear system in form of (2) and the sampled-data TLFCFFNN-
BiNj
-M
B,N;-Mij -Y
O
L
0
O 01
where the connection weights of the TLFCFFNN are defined Gi = NJX- . The maximum sampling period h is
as
obtained by minimizing - subject to the above LMIs. X,Y,M,j,M,j,M,Nj
h
B. Tuning of mji and bj In Theorem 1, the maximum sampling period and the connection weight gkj are determined based on the LMI approach. In this section, the connection weight mji and the bias bj will be determined. Owning to the nonlinear nature of the activation function of the TLFCFFNN, it is difficult to formulate the finding of mji and bj into an LMI problem. Instead, GA can be employed to tune the values of mji and bj by minimizing the following performance index,
.o[u(t) [J 2T J 3 Ju(t)J"
(28)
It should be noted that the values of the sampling period h and the connection weight gkj are kept constant during the turning process. As the system stability is determined by the sampling period and the connection weight gkj only, the values of mji and bj can be freely tuned to optimize the system performance. During the tuning process, mji is the gene to form the chromosome of the GA process.
C. Design Procedure The design procedure of the sampled-data TLFCFFNNbased controller is given as follows. Step I) Obtain the model of the nonlinear plant in the form of (2).
Step II) Determine the number of hidden node nh and the activation functions for the sampled-data TLFCFFNN-based controller in the form of (6). Step III) Determine the maximum sampling period, hm., and the connection weight gkj by Theorem 1. Choose a constant sampling period 0 < h < hma.x Step IV) Under the chosen sampling period h and the connection weight gkj, obtain the connection weight mji and the bias bj by the GA process. Realize the sampled-data TLFCFFNN-based controller based on the determined h, gkj, mji and bj. IV APPLICATION EXAMPLE
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The proposed sampled-data TLFCFFNN-based controller will be employed to stabilize an inverted pendulum subject to parameter uncertainties. The objective is to drive all the system states of the inverted pendulum to zero at the steady state.
pendulum. From (6), the sampled-data TLFCFFNN-based controller is given by, 4 u(t) = E2mj(x(t,))G jx(t7) , t, < t < t)
The system behaviour of the inverted Step I) pendulum is described by the following dynamic equation. g sin(9(t)) - ampL.9(t)2 sin(20(t)) /2 - a cos(9(t))u(t) 4L/3 - ampLcos2(6(t))
transfer function, i.e, tf (
(29)
6(t) is the angular displacement of the pendulum, g = 9.8m/s2 is the acceleration due to gravity, mp E [mp mp ] = [2 5]kg is the mass of the pendulum, M, E [Memin Mc, ] = [30 35]kg is the mass of the cart, a = 11(mp + MJ), 2L = 1 m is the length of the pendulum, and u(t) is the where
force applied to the cart. mp and MC contain parameter uncertainties of the system. The inverted pendulum subject to parameter uncertainties can be represented by the following model, 4 x(t) = E w; (Aix(t) + Biu(t)) (30) ,=,
XI (t)E
[x
xl ]
= [-
f2(x(t))
Al =A2=
0
B, =B3=Lf
fl
=
-
1
4L13-am Lcos( 2(X) 4L/3-amALcos2 (X1
B2= B4
16.4640, f2.
(t)E
ampLcos2(xl (t)
=
f[fj
XI
;
(t))
f2.
;
= -0.0492;
=E WMI~(AZx(t)))XflM, (f2(X(t)))) (xW wi~~~~~~It 4
flM(l,(f(x(t)))=- -f(x(0t)+f, M
(A, (X(t)))
=
~(rx(t))) =
flMw(W2(x(t))) =
1-fM (t ) (X(t))) -f2
1-
((t
)) +
MI(
,W for for
for
2(x(t))) for
K
=
=
1,
2;
=
3,
4;
3
and
1,
+
Step III) The maximum sampling period, hm.,, is found to be 0.0284s based on Theorem 1. A constant sampling period 0 < h = 0.02 < h,. is chosen for the sampled-data TLFCFFNN-based controller. MATALB LMI tool box is employed to solve the solution of the GEVP in Theorem 1 numerically. The connection weights, G1 = N1X-', are obtained as GI = [g1j1 g2,1] = [1358.9268 347.8515], G2 = [g1,2 g22] = G3 = [g113 g2,3] = [1319.7667 337.4474] and G4 = [g114 g924] = [1291.0126 329.7995]. It can be concluded from Theorem 1 that the optimized sampled-data TLFCFFNN-based control system is asymptotically stable. Step IV) The parameters mji and bj of the activation function
~~~~~~~~~1 can be obtained by
2
(ty) +bj?J= 2mj.,rj , (ty)+bj) tfQlmjjxi(i=l~~~~~~~~~~Y )
)) ( |~~~~(t) )
=11.3533 and
-0.0192 and
=
optimizing the following performance index using improved GA [16] under m, = 2kg and Mp = 30kg.
LX2,X2,,,n
0 1 fi0
A3=A4
and
and
x
3
3
4L/3-
and
and
-
mjjxj (t7) + bj
l+e4i='
x(t) = [x1 (t) x2 ()] = [9(t) @(t)]T
where
j=I
where the logarithmic sigmoid function is employed as the
9=
2, 4.
Step II) A sampled-data TLFCFFNN-based ci ontroller with four hidden nodes is employed to handle the inverted
2J JX[(t) dt (31) The values of the connection weight and bias, mji and bj, before and after GA process are listed in Table I. The obtained sampled-data TLFCFFNN-based controller is employed to handle the inverted pendulum subject to parameter uncertainties. Under the initial state conditions of f
[ut)
x(0) =
3
I
T
0l IT
x(0) = x)
x(0)
6J~ of6()[-
0 ]
and n
x(0) = [-
] with m, = 2kg and Mp = 30kg, the system
responses
and control signals of the sampled-data
TLFCFFNN-based control systems are shown in Fig. 2. Referring to these figures, it can be seen that the proposed sampled-data TLFCFFNN-based controller is able to stabilize the- inverted pendulum. The proposed sampled-data TLFCFFNN-based controller can be easily implemented by a microcontroller or a digital computer. V. CONCLUSION A sampled-data TLFCFFNN-based controller, which is formed by a sampler, a TLFCFFNN and a ZOH unit, has been proposed for continuous-time nonlinear systems. Based on the Lyapunov-based approach, the stability of the sampled-data TLFCFFNN-based control systems has been investigated. Stability conditions have been derived to guarantee the stability of the sampled-data TLFCFFNN-
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based control systems respectively. The findings of the maximum sampling period and the network connection weights, and the optimization of system performance have been formulated as generalized eigenvalue and genetic algorithm minimization problems. An application example on stabilizing an inverted pendulum subject to parameter uncertainties has been given to illustrate the design procedure and the effectiveness of the proposed approach.
1.5
O
REFERENCES
[1] B. Widrow and M.A. Lehr, "30 years of adaptive neural networks: Perceptron, madaline, and backpropagation," Proceedings of the IEEE, vol. 78, no. 9, Sept. 1990, pp. 1415-1442. [2] G.P. Liu, V. Kadirkamanathan, and S.A. Billings, "Variable neural networks for adaptive control of nonlinear systems," IEEE Transactions on System, Man, and Cybernetic - Part C: Applications and Reviews, vol. 29, no. 1, pp. 3443, Feb. 1999. [3] H.D. Patinto, R. Carelli, and B.R. Kuchen, "Neural network for advanced control of robot manipulators," IEEE Transactions on Neural Networks, vol. 13, no.2, pp. 343-354, March 2002. [4] R.M. Sanner and J.J.E. Slotine, "Gaussian networks for direct adaptive control," IEEE Transactions on Neural Networks, vol. 3, no. 6, pp. 837-863, Nov. 1992. [5] R. Carelli, E.F. Camacho and D. Patifio, "A neural network based feedforward adaptive controller for robots," IEEE Transactions on Systems, Man, and Cybernetics, vol. 25, no. 9, pp. 1281-1288, Sept. 1995. [6] C.L. Hwang, "Neural-network-based variable structure control of electrohydralic servosystems subject to huge uncertainties without persistent excitation," IEEE Transactions on Mechatronics, vol. 4, no. 1, pp. 50-59, March 1999. [7] S. Lin and A.A. Goldenberg, "Neural-network control of mobile manipulators," IEEE Transactions on Neural Networks, vol. 12, no. 5, pp. 1121-1133, Sept. 2001. [8] C.L. Lin, "Control of perturbed systems using neural networks," IEEE Transactions on Neural Networks, vol. 9, no. 5, pp. 1046-1050, Sept. 1998. [9] K. Tanaka, "Stability and stabilizability of fuzzy-neural-linear control systems," IEEE Transactions on Fuzzy Systems, vol. 3, no. 4, pp. 438447, Nov. 1995. [10] K. Tanaka, "An approach to stability criteria of neural-network control systems," IEEE Transactions on Neural Network, vol. 7, no. 3, pp. 629-642, May 1996. [11] J.D. Hwang and F.H. Hsiao, "Stability analysis of neural network interconnected systems," IEEE Transactions on Neural Networks, vol. 14, no. 1, pp. 201-208, Jan. 2003. [12] S. Boyd, L. El. Ghaoui, E. Feron, and V. Balakrishnan, Linear matrix inequalities in system and control theory. SLAM, Philadelphia, 1994. [13] Z. Michalewicz, Genetic Algorithm + Data Structures = Evolution Programs, 2nd Ed. Springer-Verlag, 1994. [14] F.M. Ham and I. Kostanic, Principles of Neurocomputing for Science & Engineering, McGraw Hill, 2001. [15] B.D.O. Anderson and J.B. Moore, Optimal Control: Linear Quadratic Methods. Prentice Hall, 1990. [16] F.H.F. Leung, H.K. Lam, S.H. Uing and P.K.S. Tam, "Tuning of the structure and parameters of neural network using an improved genetic algorithm," IEEE Trans. Neural Networks, vol. 14, no. 1, pp. 79-88, Jan. 2003.
-1.5I 0 0.2 0.4 0.6 0.8
1 1.2 Time (sec)
1.6
1.8
2
1.4
1.6
1.8
2
Fig. 2(a). xl(t).
Fig. 2(b). x2(t).
0
0.2
0.4
0.6
0.8
1 1.2 Tim (s.c)
Fig. 2(c). u(t) Fig. 2. System responses and control signals of the inverted pendulum with the proposed sampled-data TLFCFFNN-based controller for mp = 2kg and
MC = 30kg.
TABLE 1. THE VALUES OF THE CONNECTION WEIGHT AND BIAS, M,,, AND Bi,
Parameter ml X Ml.2
bi
M2.1
M2.2
Continuous-time |
1.4
b2
tINonlinear Plant |I*xt
M31
M3.2 b3
M4_ 1
1742 _
_
BEFORE AND AFTFR THE (',A PRAFq.C1
Initial parameter values -0.0591
0.5511 --0.2765 --0.9863 0.5118 -0.3234 0.3058 0.1582 -0.7899 _-0.6254 -0.7569 0.4423
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Parameter values after GA process -0.7283 -0.7605 -0.2638 0.8534 -0.8836 0.9208
0.9816 0.9656 0.8918 0.9972 0.9993 0.8240
